Invariant rational functions under rational transformations

Abstract

Let $X$ be an algebraic variety equipped with a dominant rational self-map $\ddot{I}•:X\to X$. A new quantity measuring the interaction of $(X,\ddot{I}•)$ with trivial dynamical systems is introduced; the stabilised algebraic dimension of $(X,\ddot{I}•)$ captures the maximum number of new algebraically independent invariant rational functions on the cartesian product of $(X,\ddot{I}•)$ and $(Y,\ddot{I}ˆ)$, as $(Y,\ddot{I}ˆ)$ ranges over all algebraic dynamical systems. It is shown that this birational invariant agrees with the maximum dimension of a dominant equivariant rational image $(X^{\prime },\ddot{I}{•}^{\prime })$ where $\ddot{I}{•}^{\prime }$ is part of an algebraic group action on $X^{\prime }$. As a consequence, it is deduced that if some cartesian power of $(X,\ddot{I}•)$ admits a nonconstant invariant rational function, then already the second cartesian power does.

20 pages, to appear in Selecta